Periodical properties of multidimensional polynomial transformations of Galois – Eisenstein ring

O. A. Kozlitin

Matematičeskie voprosy kriptografii, Tome 13 (2022) no. 1, pp. 69-99

Résumé

The paper is concerned with $m$-dimensional polynomial transformations of Galois – Eisenstein ring $R$ (that is a finite commutative local ring of principal ideals). The maximum $L_m(R)$ cycle lengths of such polynomial transformations is estimated. Under condition $p > 2$, the constraint of the function $L_m$ on the class of Galois – Eisenstein rings having a power $q_n = p^{tn}$ and nilpotency index $n$ takes the maximum value on the Galois rings.

Bibliographie

[1] Anashin V. S., “O gruppakh i koltsakh, obladayuschikh tranzitivnymi polinomami”, XVI Vsesoyuzn. algebr. konf., tezisy, v. II, 1981, 4–5

[2] Asanov M. O., Baranskii V. A., Rasin V. V., Diskretnaya matematika: grafy, matroidy, algoritmy, Uchebnoe posobie, 2-e izd., ispr. i dop., Lan, Sankt-Peterburg–Moskva–Krasnodar, 2010, 368 pp.

[3] Elizarov V. P., Konechnye koltsa. Osnovy teorii, Gelios-ARV, M., 2006, 304 pp.

[4] Viktorenkov V. E., “Orgraf polinomialnogo preobrazovaniya nad kommutativnym lokalnym koltsom”, Obozr. prikl. i promyshl. matem., 7:2 (2000), 327

[5] Viktorenkov V. E., “O stroenii orgrafov polinomialnykh preobrazovanii nad konechnymi kommutativnymi koltsami s edinitsei”, Diskretnaya matematika, 29:3 (2017), 3–23

[6] Glukhov M. M., Elizarov V. P., Nechaev A. A., Algebra, Uchebnik, 2-e izd., ispr. i dop., Lan, SPb., 2015, 608 pp.

[7] Ermilov D. M., “Algoritm postroeniya sistemy predstavitelei tsiklov maksimalnoi dliny polinomialnykh podstanovok nad koltsom Galua”, Prikl. diskret. matem. Prilozhenie, 7 (2014), 64–67

[8] Ermilov D. M., Kozlitin O. A., “Tsiklovaya struktura polinomialnogo generatora nad koltsom Galua”, Matematicheskie voprosy kriptografii, 4:1 (2013), 27–57 | Zbl

[9] Ermilov D. M., Kozlitin O. A., “O stroenii grafa polinomialnogo preobrazovaniya koltsa Galua”, Matematicheskie voprosy kriptografii, 2015, no. 3, 47–73 | Zbl

[10] Kozlitin O. A., “Otsenka dliny maksimalnogo tsikla v grafe polinomialnogo preobrazovaniya koltsa Galua – Eizenshteina”, Diskretnaya matematika, 29:4 (2017), 41–58

[11] Kozlitin O. A., “Periodicheskie svoistva mnogomernogo polinomialnogo generatora nad koltsom Galua, I”, Matematicheskie voprosy kriptografii, 9:3 (2018), 61–98 | MR | Zbl

[12] Kozlitin O. A., “Periodicheskie svoistva mnogomernogo polinomialnogo generatora nad koltsom Galua, II”, Matematicheskie voprosy kriptografii, 11:1 (2020), 63–100 | MR | Zbl

[13] Kozlitin O. A., “Periodicheskie svoistva mnogomernogo polinomialnogo generatora nad koltsom Galua, III”, Matematicheskie voprosy kriptografii, 11:4 (2020), 49–76 | MR | Zbl

[14] Kuzmin A. S., Kurakin V. L., Nechaev A. A., “Psevdosluchainye i polilineinye posledovatelnosti”, Trudy po diskretnoi matematike, 1 (1997), 139–202 | Zbl

[15] Larin M. V., “Tranzitivnye polinomialnye preobrazovaniya kolets vychetov”, Diskretnaya matematika, 14:2 (2002), 20–32 | Zbl

[16] Nechaev A. A., “Konechnye koltsa glavnykh idealov”, Matem. sb., 91:3 (1973), 350–366

[17] Nechaev A. A., “O stroenii konechnykh kommutativnykh kolets s edinitsei”, Matem. zam., 10:6 (1971), 679–688 | Zbl

[18] Nechaev A. A., “Polinomialnye preobrazovaniya konechnykh kommutativnykh lokalnykh kolets glavnykh idealov”, Matem. zam., 27:6 (1980), 885–897 | MR | Zbl

[19] Nechaev A. A., “Tsiklovye tipy lineinykh podstanovok nad konechnymi kommutativnymi koltsami”, Matem. sb., 184:3 (1993), 21–56 | Zbl

[20] Fisher B., “A note on Hensel's lemma in several variables”, Proc. Amer. Math. Soc., 125 (1997), 3185–3189 | DOI | MR | Zbl